Explicit Computation of Stabilizing Feedback Control Gains Using Polyhedral Lyapunov Functions

The explicit computation of stabilizing Static Output Feedback (SOF) and State Feedback (StF) control gains is treated in this work by considering polyhedral Lyapunov functions (PLF). More specifically, the associated nonlinear algebraic necessary and sufficient conditions that prove closed-loop stability are exploited to propose bilinear optimization problems that allow to explicitly design the control gains. The objective function considered here in allows to weigh the speed of states convergence and the control effort or the size of a region of admissible initial conditions in case of constrained control inputs. Furthermore, the proposed design for optimization problems can be implemented by using an existing non-linear optimization solver that shows to be specially efficient to deal with the considered bilinearities. A numerical example is presented to show the effectiveness of our proposal for both SOF and StF designs. Some comments about future research directions using the considered framework are also provided.